Question

The total number of terms in the expansion of \( (x + y)^{100} + (x - y)^{100} \):

• A. 49
• B. 50
• C. 51
• D. 99

Hint:

In \( (x + y)^n + (x - y)^n \), odd powers of \( y \) cancel and even powers double.

Answer 1

The Correct Option is C

The given expression is \( (x + y)^{100} + (x - y)^{100} \). To find the total number of terms in the expansion of this expression, we will first analyze the individual expansions of \( (x + y)^{100} \) and \( (x - y)^{100} \), and then combine them.

Step 1: Expansion of \( (x + y)^{100} \)

The binomial expansion of \( (x + y)^{100} \) is given by:

\[ (x + y)^{100} = \sum_{k=0}^{100} \binom{100}{k} x^{100-k} y^k \]

This expansion has \( 101 \) terms because the index \( k \) ranges from 0 to 100, inclusive. So, the expansion of \( (x + y)^{100} \) contains 101 terms.

Step 2: Expansion of \( (x - y)^{100} \)

The binomial expansion of \( (x - y)^{100} \) is given by:

\[ (x - y)^{100} = \sum_{k=0}^{100} \binom{100}{k} x^{100-k} (-y)^k = \sum_{k=0}^{100} \binom{100}{k} (-1)^k x^{100-k} y^k \]

Like the first expansion, this also has \( 101 \) terms. However, the signs of the terms alternate depending on whether \( k \) is even or odd due to the factor \( (-1)^k \). So, the expansion of \( (x - y)^{100} \) also contains 101 terms.

Step 3: Combining the Expansions

Now, consider the sum of the two expansions:

\[ (x + y)^{100} + (x - y)^{100} \] When we add these two expansions, terms where the powers of \( y \) are odd will cancel each other out, because \( (-y)^k \) will have the opposite sign of \( y^k \) when \( k \) is odd. On the other hand, terms where \( k \) is even will add up, because both expansions will have the same sign for those terms.

 

Thus, only the terms where \( k \) is even will remain after addition. The even values of \( k \) range from 0 to 100, inclusive. These values are \( k = 0, 2, 4, \dots, 100 \), which gives a total of 51 even values.

Final Answer:

The total number of terms in the expansion of \( (x + y)^{100} + (x - y)^{100} \) is 51, because only the terms with even values of \( k \) survive after adding the two expansions